Problem: Aleksandra started studying how the number of branches on her tree grows over time. Every $1.5$ years, the number of branches increases by an addition of $\dfrac{2}{7}$ of the total number of branches. The number of branches can be modeled by a function, $N$, which depends on the amount of time, $t$ (in years). When Aleksandra began the study, her tree had $52$ branches. Write a function that models the number of branches $t$ years since Aleksandra began studying her tree. $N(t) = $
Answer: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of branches is $52$, and every $1.5$ years, the number of branches increases by an addition of $\dfrac{2}{7}$ of the total number of branches. Note that increasing by $\dfrac{2}{7}$ is the same as being multiplied by $\dfrac{9}{7}$. [Why?] This means that the initial quantity is $A=52$ and the factor is $B=\dfrac{9}{7}$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac{9}{7}$ every $1.5$ years. Finding the expression in the exponent We know that the number of branches is multiplied by $\dfrac{9}{7}$ every $1.5$ years. This means that each time $t$ increases by $1.5$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{1.5}$. When the initial measurement is made, the number of branches hasn't changed. So $N(0) = 52$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{1.5}$. Summary We found that the following function models the number of branches $t$ years since Aleksandra began studying her tree. N ( t ) = 52 ⋅ ( 9 7 ) t 1.5 N(t)=52\cdot \left(\dfrac{9}{7}\right)\^{ \frac{t}{1.5}}